Functional analysis

Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space is a linear functional on so that for all positive elements that is it holds that In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When is a complex vector space, it is assumed that for all is real. As in the case when is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace and the partial order does not extend to all of in which case the positive elements of are the positive elements of by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any equal to for some to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products. (Wikipedia).

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Support (mathematics) | Cauchy–Schwarz inequality | Functional analysis | Riesz–Markov–Kakutani representation theorem | Topological vector lattice | Absorbing set | Ordered topological vector space | Hausdorff space | C*-algebra | Normal cone (functional analysis) | Sesquilinear form | Borel regular measure | Riesz space | Fréchet space | Mathematics | Square matrix | Ordered vector space | Bornological space | Compact space | Operator system | Complex number